The Implications of the No-Free-Lunch Theorems for Meta-induction

TL;DR

This paper discusses the implications of no-free-lunch theorems for meta-induction, emphasizing the limitations of NFL theorems and highlighting the importance of free lunch theorems and statistical correlations in induction.

Contribution

It clarifies the scope of NFL theorems, introduces the concept of free lunch theorems, and critiques the uniform prior advocated by Schurz based on empirical evidence.

Findings

NFL theorems do not only concern uniform priors

Free lunch theorems relate to correlations among generalization errors

Schurz's uniform prior is contradicted by statistical physics experiments

Abstract

The important recent book by G. Schurz appreciates that the no-free-lunch theorems (NFL) have major implications for the problem of (meta) induction. Here I review the NFL theorems, emphasizing that they do not only concern the case where there is a uniform prior -- they prove that there are "as many priors" (loosely speaking) for which any induction algorithm $A$ out-generalizes some induction algorithm $B$ as vice-versa. Importantly though, in addition to the NFL theorems, there are many {free lunch} theorems. In particular, the NFL theorems can only be used to compare the {marginal} expected performance of an induction algorithm $A$ with the marginal expected performance of an induction algorithm $B$. There is a rich set of free lunches which instead concern the statistical correlations among the generalization errors of induction algorithms. As I describe, the meta-induction algorithms that Schurz advocate as a "solution to Hume's problem" are just an example of such a free lunch based on correlations among the generalization errors of induction algorithms. I end by pointing out that the prior that Schurz advocates, which is uniform over bit frequencies rather than bit patterns, is contradicted by thousands of experiments in statistical physics and by the great success of the maximum entropy procedure in inductive inference.